Estimation of the remainder of a cubature formula on a Chebyshev grid

V. A. Abilov ; M. K. Kerimov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 8, pp. 1373-1377

Résumé

Let $C(Q)$ denote the space of continuous functions $f(x,y)$ in the square $Q=[-1,1]\times[-1,1]$ with the norm \begin{equation} \| f\|=\max(|(f(x,y)|), \quad (x,y)\in Q \end{equation} On a Chebyshev grid, a cubature formula of the form \begin{eqnarray} \int_{-1}^1\int_{-1}^1\frac{1}{\sqrt{(1-x^2)(1-y^2)}}f(x,y)dxdy= \\ \frac{\pi^2}{mn}\sum_{i=1}^n\sum_{j=1}^mf\big (\cos\frac{2i-1}{2n}\pi,cos\frac{2j-1}{2m}\pi\big )+R_{m,n}(f) \end{eqnarray} is considered in some class $H(r_1,r_2)$ of functions $f\in C(Q)$, defined by a generalized shift operator. The remainder $R_{m,n}(f)$ is proved to satisfy the estimate: $$ \sup_{f\in H(r_1,r_2)}| R_{m,n}(f) |=O(n^{-r_1+1}+m^{-r_2+1}) $$ where $r_1,r_2>1,\lambda^{-1}\leq n/m\leq\lambda,\lambda>0$; and the constant in $O(1)$, depends on $\lambda$. Библ. 4. Ключевые слова: кубатурная формула, чебышевская сетка, оценка остаточного члена.